Math Problem Statement
Solution
We are given two points with coordinates and , and the line joining these points is perpendicular to the line .
To find the value of , let's go through the following steps:
Step 1: Find the slope of the given line .
We need to rewrite this equation in slope-intercept form , where is the slope:
So, the slope of the given line is .
Step 2: Find the slope of the line joining the points and .
The formula for the slope of a line between two points and is:
Substitute the coordinates of the points:
Step 3: Use the perpendicularity condition.
If two lines are perpendicular, the product of their slopes is . Let the slope of the given line be and the slope of the line joining the points be . Set up the equation:
Step 4: Solve for .
Now, solve the equation:
Multiply both sides by to eliminate the denominator:
Expand both sides:
Move all terms involving to one side and constants to the other side:
Solve for :
Final Answer:
The value of is , which corresponds to option C.
Would you like further details on this process or additional explanations?
Here are five related questions you might want to explore:
- How do you find the equation of a line perpendicular to a given one?
- What is the general condition for two lines to be perpendicular?
- How do you calculate the midpoint between two points in coordinate geometry?
- Can you find the equation of a line passing through two points?
- What happens when two lines are parallel?
Tip: When working with slopes, always remember that perpendicular slopes are negative reciprocals of each other!
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Slopes of Perpendicular Lines
Linear Equations
Formulas
Slope formula: (y2 - y1) / (x2 - x1)
Perpendicular slope condition: m1 * m2 = -1
Theorems
Perpendicular Slopes Theorem
Suitable Grade Level
Grades 10-12
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